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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V7">commutative</a> <a href="vectsp_2.html#V2">domRing-like</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:104"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="quofield.html#R3" target="_self">has_Field_of_Quotients_Pair</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="quofield.html#R3" target="_self">has_Field_of_Quotients_Pair</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:104"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="quofield.html#V3" target="_self">RingMonomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#D29" target="_self">Def29</a></i>;<br/>
<a NAME="E3:104"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="quofield.html#V3" target="_self">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:104"><i><font color="Green">E82</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self">Def29</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:104"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:104"><i><font color="Green">E82</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self">Def29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:104"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:104"><i><font color="Green">E82</font></i></a>, <a class="txt" href="quofield.html#E2:104"><i><font color="Green">E83</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self">Def29</a></i>;<br/>
<a NAME="E8:104"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:104"><i><font color="Green">E85</font></i></a>, <a class="txt" href="quofield.html#E7:104"><i><font color="Green">E86</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E9:104"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:104"><i><font color="Green">E85</font></i></a>, <a class="txt" href="quofield.html#E7:104"><i><font color="Green">E86</font></i></a>, <a class="ref" href="quofield.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E10:104"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:104"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:104"><i><font color="Green">E82</font></i></a>, <a class="txt" href="quofield.html#E2:104"><i><font color="Green">E83</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self">Def29</a></i>;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="quofield.html#E8:104"><i><font color="Green">E87</font></i></a>, <a class="txt" href="quofield.html#E9:104"><i><font color="Green">E88</font></i></a>, <a class="txt" href="quofield.html#E12:104"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="quofield.html#E10:104"><i><font color="Green">E89</font></i></a>, <a class="txt" href="quofield.html#E12:104"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>

;<br/>
<a NAME="E14:104"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#T37">FUNCT_2:37</a></i>;<br/>
<a NAME="E15:104"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V3" target="_self">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:104"><i><font color="Green">E85</font></i></a>, <a class="ref" href="quofield.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E16:104"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:104"><i><font color="Green">E85</font></i></a>, <a class="txt" href="quofield.html#E7:104"><i><font color="Green">E86</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E17:104"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:104"><i><font color="Green">E85</font></i></a>, <a class="txt" href="quofield.html#E7:104"><i><font color="Green">E86</font></i></a>, <a class="ref" href="quofield.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E18:104"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E20:104"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:104"><i><font color="Green">E82</font></i></a>, <a class="txt" href="quofield.html#E3:104"><i><font color="Green">E84</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self">Def29</a></i>;<br/>
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="quofield.html#E16:104"><i><font color="Green">E92</font></i></a>, <a class="txt" href="quofield.html#E17:104"><i><font color="Green">E93</font></i></a>, <a class="txt" href="quofield.html#E20:104"><i><font color="Green">E95</font></i></a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="quofield.html#E18:104"><i><font color="Green">E94</font></i></a>, <a class="txt" href="quofield.html#E20:104"><i><font color="Green">E95</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>

;<br/>
<a NAME="E22:104"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T24">FUNCT_2:24</a></i>;<br/>
<a NAME="E23:104"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V2" target="_self">RingEpimorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:104"><i><font color="Green">E85</font></i></a>, <a class="ref" href="quofield.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E24:104"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V4" target="_self">RingIsomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E15:104"><i><font color="Green">E91</font></i></a>, <a class="ref" href="quofield.html#D24" target="_self">Def24</a></i>;<br/>
<b>hence </b><a NAME="E25:104"/>
<font color="Maroon">c<sub>2</sub></font> <a href="quofield.html#R2" target="_self">is_ringisomorph_to</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="quofield.html#D26" target="_self">Def26</a></i>;<br/>


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